Depends on the signature flag in the transaction. If no flag is specified, the signature is calculated based on the entire transaction. *
That means that for a standard transaction, a change in the order of either the inputs or the outputs will break the signatures.
See the hashtype values
here for more information.
And yes, changing the order of the transactions in the block will change the merkle root, which will change the block header and invalidate the hash.
* Technical detail: the signature fields in the inputs are deleted for the purposes of calculating the signatures.